Q. 44 - Calculus

Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\mathrm{d}x={\mathrm{e}}^{\sqrt{\mathrm{x}}}+\mathrm{C}$.

Step 1. Given Information

Solving the given integrals. 

$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\mathrm{d}x$

Step 2. Solving the given integral using substitution method.

$$\begin{gathered} u=\sqrt{x} \\ u=x^{1/2} \\ \frac{du}{dx}=\frac{1}{2}{x}^{-1/2} \\ \frac{du}{dx}=\frac{1}{2\sqrt{x}} \\ 2du=\frac{1}{\sqrt{x}}dx \end{gathered}$$

Step 3. This substitution changes the integral into

$$\begin{gathered} \int \frac{e^{\sqrt{x}}}{\sqrt{x}}\mathrm{d}x=\int e^u\mathrm{du} \\ \int \frac{{\mathrm{e}}^{\sqrt{\mathrm{x}}}}{\sqrt{\mathrm{x}}}\mathrm{dx}={\mathrm{e}}^{\mathrm{u}}+\mathrm{C} \\ \int \frac{{\mathrm{e}}^{\sqrt{\mathrm{x}}}}{\sqrt{\mathrm{x}}}\mathrm{dx}={\mathrm{e}}^{\sqrt{\mathrm{x}}}+\mathrm{C} \end{gathered}$$